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Full text of "Treatise On Analysis Vol-Ii"

14   INTEGRATION WITH RESPECT TO A POSITIVE MEASURE        185

(13.7.7). But, by (13.8.4), if N'=>N is the intersection of the UM, then
inf \g<p\jn d\i = \g(pw dp = 0; and since v(N) ^ v(Uw) for all n, we have

v(N) = 0.

Now let N be any /^-negligible set, and (KJ a denumerahle covering of X
by compact sets. Then the sets N n Kn are v-negligible, by what has just
been proved, and hence so is their union N.

(13.14.1.4)    Suppose now that K = Supp(/) is compact and that f\ K is
continuous, with values in R (and therefore bounded (3.17.10)). Then there
exists a decreasing sequence (Un) of relatively compact neighborhoods of K
such that K = p) Un (3.18.2); also, by virtue of the Tietze-Urysohn theorem

n

(4.5.1), there exists for each n a function / e ^TR(X), with support con-
tained in Uw, which extends/and is such that ||/J = ||/||. Hence we have

\fn dv = \fng d\i for all n. Bearing in mind (13.13.1), it follows from (13.8.4)
that/is v-integrable and/# is /t-integrable, and that jfdv = \fg dp.

(13.14.1.5)    The set A of points x e X such that g(x) = 6 is ^-negligible.

Since g is ju-measurable (13.13.1), A is//-measurable (13.9.9), and hence is
the union of a sequence (Kn) of compact sets and a /(-negligible set N. By

virtue of (13.14.1.4) applied to/= (pKn, we have v(Kw) = j gcpKn dju = 0, and
v(N) = 0 by (13.14.1.3); hence v(A) = 0.

(13.14.1.6)    Consider now the case where Supp(/) = K is compact and/| K
is lower semicontinuous on K. Then it follows from (12.7.8) that there exists
an increasing sequence of finite real-valued functions un which are continuous
and ^ 0 on K and such that/|K = sup u. Let/n be the function which is

n

equal to un on K and zero on X - K, so that/= sup/,. By virtue of (13.5.7)

n

and (13.14.1.4), we have

fdv = sup     fn dv = sup     fng dp, = \ fg d^
since fg is equivalent (with respect to /i) to sup/,0, by virtue of the convention

n

about products and the fact that g is finite almost everywhere with respect

tO fJL.he difference between 0